Z Transform Filters Introduction.

Cuthbert Nyack
In the same way that analog filters can be designed by placement of poles and zeros in the s plane, then digital filters can also be designed by suitable location of poles and/or zeros in the Z plane. Digital filters are often categorised as finite impulse response(FIR) and infinite impulse response(IIR). Finite impulse response filters can be implemented with zeros and IIR filters with poles and zeros. Software packages are available for the design of FIR filters and IIR filters can be designed by sampled approximations to classical analog responses.
Here we look at some simple filters to illustrate some of their properties and some of the basic concepts used in the design, eg simple pole-zero configurations, some named filters (allpass, linear phase, comd etc), sampled approximations to analog designs, FFT and frequency sampling filters.
Definitions of Low pass, High pass and Band pass filters, indicated above, have to be modified when referring to digital filters because of the periodicity of their responses. LP, HP and BP filters now refer to the frequency interval from 0 to half the sampling frequency instead from 0 to infinity. The spectra of digital filters are periodic with a period of ws.
If a filter has 2 CC zeros shown by the 2 black o's above and 2 CC poles shown by the black x's above then the magnitude and phase of the response at w is given by Mag = (a1 ´ a2)/ (b1 ´ b2) and Phase = (q1 + q2) - (q3 + q4).
a1, a2 are the distances from the zeros to the point which subtends wT at the center. (T = 2p/ws).